^{1,a)}and Gerhard Gompper

^{1,b)}

### Abstract

The dynamics of membranes is studied on the basis of a particle-based meshless surface model, which was introduced earlier [Phys. Rev. E73, 021903 (2006)]. The model describes fluid membranes with bending energy and—in the case of membranes with boundaries—line tension. The effects of hydrodynamic interactions are investigated by comparing Brownian dynamics with a particle-based mesoscale solvent simulation (multiparticle collision dynamics). Particles self-assemble into vesicles via disk-shaped membrane patches. The time evolution of assembly is found to consist of three steps: particle assembly into discoidal clusters, aggregation of clusters into larger membrane patches, and finally vesicle formation. The time dependence of the cluster distribution and the mean cluster size is evaluated and compared with the predictions of Smoluchowski rate equations. On the other hand, when the line tension is suddenly decreased (or the temperature is increased), vesicles dissolve via pore formation in the membrane. Hydrodynamic interactions are found to speed up the dynamics in both cases. Furthermore, hydrodynamics makes vesicle more spherical in the membrane-closure process.

Stimulating discussions with S. Egelhaaf (Düsseldorf) are gratefully acknowledged.

I. INTRODUCTION

II. MODEL AND METHOD

A. Curvature potential

B. Attractive and repulsive potentials

C. Simulation methods: BD and MPC

D. Membrane properties

III. FORMATION OF DISKLIKE MICELLES AND VESICLES

A. Self-assembly

B. Growth of disklike micelles

C. Vesicle closure

IV. VESICLEDISSOLUTION

V. SUMMARY

## Figures

(Color online) Membrane viscosity as a function of for two values of the surface tension, and , as indicated.

(Color online) Membrane viscosity as a function of for two values of the surface tension, and , as indicated.

Snapshots of different stages of the self-assembly process in a BD simulation at the density of membrane particles , with and . (a) , (b) , and (c) .

Snapshots of different stages of the self-assembly process in a BD simulation at the density of membrane particles , with and . (a) , (b) , and (c) .

(Color online) Time development of the mean cluster size for self-assembly at . The solid lines represent data for , 0.0076, and 0.015 (from bottom to top) without hydrodynamic interactions (BD). The dashed lines represent data for with hydrodynamic interactions (MPC). Error bars are shown at several data points.

(Color online) Time development of the mean cluster size for self-assembly at . The solid lines represent data for , 0.0076, and 0.015 (from bottom to top) without hydrodynamic interactions (BD). The dashed lines represent data for with hydrodynamic interactions (MPC). Error bars are shown at several data points.

(Color online) Time development of the average number of clusters for , , and , in a system without hydrodynamic interactions (BD). The solid and dashed lines represent the number of vesicles and the other clusters, respectively. Error bars are shown at several data points.

(Color online) Time development of the average number of clusters for , , and , in a system without hydrodynamic interactions (BD). The solid and dashed lines represent the number of vesicles and the other clusters, respectively. Error bars are shown at several data points.

(Color online) (a) Time development of the average and the standard deviation of the cluster-size distribution. The dashed lines represent the simulation data for and with (BD) or 0.0076 (MPC). The top two lines represent the BD data for (upper line) and (lower line); the bottom two lines represent the MPC data. The solid lines are calculated from Eq. (15) with the kernel (17) for nonhydrodynamic (“hy−,” ) and the kernel (16) for hydrodynamic diffusion (“hy+,” ), respectively. The data with hydrodynamics are shifted by a factor of 6 horizontally to avoid an overlap of lines. (b) Time development of the cluster density obtained from BD simulations for the various sizes , as indicated. (c) Cluster-size distributions obtained from BD simulations at various times , as indicated. In the histogram for , data for are smoothed by averaging over some range of neighboring cluster sizes. The solid lines in (b) and (c) represent the cluster densities and size distribution calculated from Eq. (15) with the simulated distribution at as initial condition.

(Color online) (a) Time development of the average and the standard deviation of the cluster-size distribution. The dashed lines represent the simulation data for and with (BD) or 0.0076 (MPC). The top two lines represent the BD data for (upper line) and (lower line); the bottom two lines represent the MPC data. The solid lines are calculated from Eq. (15) with the kernel (17) for nonhydrodynamic (“hy−,” ) and the kernel (16) for hydrodynamic diffusion (“hy+,” ), respectively. The data with hydrodynamics are shifted by a factor of 6 horizontally to avoid an overlap of lines. (b) Time development of the cluster density obtained from BD simulations for the various sizes , as indicated. (c) Cluster-size distributions obtained from BD simulations at various times , as indicated. In the histogram for , data for are smoothed by averaging over some range of neighboring cluster sizes. The solid lines in (b) and (c) represent the cluster densities and size distribution calculated from Eq. (15) with the simulated distribution at as initial condition.

Snapshot of the S-shaped conformation of a closing membrane for , , and .

Snapshot of the S-shaped conformation of a closing membrane for , , and .

(Color online) Time development of the radius of gyration for the closure of a vesicle from a flat membrane at , , and , as obtained from BD simulations. The lines represent three examples with different realizations of the thermal white noise.

(Color online) Time development of the radius of gyration for the closure of a vesicle from a flat membrane at , , and , as obtained from BD simulations. The lines represent three examples with different realizations of the thermal white noise.

(Color online) Time development of (a) the radius of gyration , (b) the asphericity , and (c) the attractive energy for vesicle closure with and . The lines represent data of BD for , 4, 5, and 6 (from bottom to top; brown, green, red, and blue). The reference time is set at ; time is rescaled by the dimensionless line tension to reduce the difference in time scales. Error bars are shown at several data points.

(Color online) Time development of (a) the radius of gyration , (b) the asphericity , and (c) the attractive energy for vesicle closure with and . The lines represent data of BD for , 4, 5, and 6 (from bottom to top; brown, green, red, and blue). The reference time is set at ; time is rescaled by the dimensionless line tension to reduce the difference in time scales. Error bars are shown at several data points.

(Color online) Relation of the average asphericity to the normalized radius of gyration during vesicle closure, for membranes of size . (a) Dependence on without hydrodynamic interactions (BD). The solid lines represent data for with , 3, and 6 (from bottom to top; brown, red, and blue). The dashed lines represent data for with , 4, and 6 (from bottom to top; red, green, and blue). [(b) and (c)] Effect of hydrodynamic interactions for . The solid lines (blue) represent the BD results. The dashed (green) and dotted (red) lines represent the MPC data with and 1.3, respectively. In (c), the spherical-cap result, Eq. (21), is given by the dashed line (black) marked “spẖcap.”

(Color online) Relation of the average asphericity to the normalized radius of gyration during vesicle closure, for membranes of size . (a) Dependence on without hydrodynamic interactions (BD). The solid lines represent data for with , 3, and 6 (from bottom to top; brown, red, and blue). The dashed lines represent data for with , 4, and 6 (from bottom to top; red, green, and blue). [(b) and (c)] Effect of hydrodynamic interactions for . The solid lines (blue) represent the BD results. The dashed (green) and dotted (red) lines represent the MPC data with and 1.3, respectively. In (c), the spherical-cap result, Eq. (21), is given by the dashed line (black) marked “spẖcap.”

(Color online) Time development of [(a) and (c)] the radius of gyration and (b) the pore angle for vesicle closure with , , and . The angle is calculated from with Eq. (19). The box size in (a) and (b) is . The lines represent MPC data with , 0.34, and 1, as indicated. The time axis is scaled with the characteristic time . The insets in (a) and (b) show the same data, but the time axis now scaled with . The dashed line in the inset of (b) shows the result of Eq. (24) with . The solid and dashed lines in (c) represent data for and , respectively.

(Color online) Time development of [(a) and (c)] the radius of gyration and (b) the pore angle for vesicle closure with , , and . The angle is calculated from with Eq. (19). The box size in (a) and (b) is . The lines represent MPC data with , 0.34, and 1, as indicated. The time axis is scaled with the characteristic time . The insets in (a) and (b) show the same data, but the time axis now scaled with . The dashed line in the inset of (b) shows the result of Eq. (24) with . The solid and dashed lines in (c) represent data for and , respectively.

(Color online) (a) Latency time and (b) closure time for a membrane of size . The full lines with symbols represent data of BD for (squares) and (circles). The dashed lines with symbols represent data of MPC for with solvent viscosities (triangles) and 1.9 (diamonds).

(Color online) (a) Latency time and (b) closure time for a membrane of size . The full lines with symbols represent data of BD for (squares) and (circles). The dashed lines with symbols represent data of MPC for with solvent viscosities (triangles) and 1.9 (diamonds).

Snapshots of the dissolution process of a vesicle in BD simulation with , , , simulation-box size , at times (a) , (b) , (c) , (d) .

Snapshots of the dissolution process of a vesicle in BD simulation with , , , simulation-box size , at times (a) , (b) , (c) , (d) .

(Color online) Time development of (a) the radius of gyration and (b) the asphericity of the largest cluster during vesicle dissolution with , , , and box size . The dashed lines represent the same data as in Fig. 12. The solid lines represent the averaged data with the reference time set at . The asphericity is only shown for the largest cluster, and as long as .

(Color online) Time development of (a) the radius of gyration and (b) the asphericity of the largest cluster during vesicle dissolution with , , , and box size . The dashed lines represent the same data as in Fig. 12. The solid lines represent the averaged data with the reference time set at . The asphericity is only shown for the largest cluster, and as long as .

(Color online) Rupture time of a vesicle with and . The circles and squares represent data of BD and MPC, respectively.

(Color online) Rupture time of a vesicle with and . The circles and squares represent data of BD and MPC, respectively.

(Color online) Time development of the average number of particles in the first and second largest clusters, for and . (a) BD results for , 1.3, and 1.4, with box size . The dashed lines are fits to Eq. (28). The inset shows the dependence of the decay rate in Eq. (28). (b) The solid lines represent results of BD simulations with , 50, 64, and 100 (from top to bottom) at . The short-dashed (red) line represents results of MPC simulations for and . The long-dashed line (black) is obtained from Eq. (30) with , , and the same decay rate as determined in (a). Error bars are shown at several data points.

(Color online) Time development of the average number of particles in the first and second largest clusters, for and . (a) BD results for , 1.3, and 1.4, with box size . The dashed lines are fits to Eq. (28). The inset shows the dependence of the decay rate in Eq. (28). (b) The solid lines represent results of BD simulations with , 50, 64, and 100 (from top to bottom) at . The short-dashed (red) line represents results of MPC simulations for and . The long-dashed line (black) is obtained from Eq. (30) with , , and the same decay rate as determined in (a). Error bars are shown at several data points.

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